Gendersleep Apnea Women Total
Gendersleep Apneayesnomen40360400women12288300total52648700here Is Da
Gendersleep Apneayesnomen40360400women12288300total52648700here Is Da
Gender Sleep Apnea? yes no Men Women Total here is data from question 1. Need #2 Performing a Chi-Square test for statistical significance of independence between two categorical variables based on Question 1 (Using the same data as Question1). a. For the Sleep apnea problem (Question 1), indicate the specific null and alternative hypotheses that will be used with a Chi-Square Test. Write the null hypothesis and alternative hypothesis for this 2 x 2 table. b. Calculate the expected counts using the method shown in the example in the online notes 11.2. Show all work. (Hint: Expected Count = n_row * n_column / n_total) c. Interpret all four expected counts in the context of the problem. (We would expect ____, if in fact there is no relationship between _____ and _____ in the population) You will have four answers--one for each expected count. d. Calculate the Chi-Square Statistic by hand using the formula e. Determine a p-value associated with the test statistics you calculated in previous question. Write down the degree of freedom and p-value from TABLE. DF = (# rows -1) (# columns - 1); p-value = f. Confirm your p-value using Minitab. Copy and Paste your output. [Hint: Minitab Express User: Statistics > Distribution Plots > Display Probability > Distribution > Chi-Square distribution > putting degree of freedom in the box > select “a specified x value” > Right tail > Putting the chi-square statistics you calculated by hand > OK] g. what is your conclusion based on your chi-square test? Reasoning?
Paper For Above instruction
Introduction
The chi-square test for independence is a fundamental statistical tool used to determine whether there is a significant association between two categorical variables. In this context, we analyze the relationship between gender and sleep apnea status based on the provided data. This examination involves formulating hypotheses, calculating expected counts, and interpreting the implications of the test results. Understanding such associations enhances our insights into demographic health patterns and can inform targeted interventions.
Hypotheses Formulation
The null hypothesis (H₀) in a chi-square test of independence posits that there is no association between the variables—in this case, gender and presence of sleep apnea. Formally, H₀ states that gender and sleep apnea status are independent in the population—that is, the distribution of sleep apnea does not differ by gender. Conversely, the alternative hypothesis (H₁) asserts that there is a dependency; gender and sleep apnea are associated, indicating the distribution of sleep apnea varies between males and females. These hypotheses set the foundation for the statistical analysis.
Calculating Expected Counts
Given the data, the total sample size (n) equals 52,648,700. The observed counts for males and females are 40,360,400 and 12,288,300 respectively, with a total of 52,648,700. The counts of sleep apnea among males and females are not explicitly specified, but assuming the problem refers to a cross-tabulation table, the expected counts are calculated based on marginal totals. Using the formula:
Expected count = (Row total * Column total) / Overall total
each cell's expected frequency is derived. For example, for males with sleep apnea, the expected count is (Total males * Total with sleep apnea) / Total sample size.
Interpreting Expected Counts in Context
The expected counts represent the frequencies we would anticipate if gender and sleep apnea status were truly independent in the population—meaning, the likelihood of having sleep apnea is unrelated to being male or female. For each cell:
- If the expected count closely matches the observed count, it suggests no association.
- Significant deviations may indicate an association, contributing to rejecting the null hypothesis.
Specifically, for the four cells (men with sleep apnea, men without, women with, women without), interpretation revolves around whether the observed data significantly depart from these expected frequencies.
Calculating Chi-Square Statistic
The chi-square statistic (χ²) quantifies the discrepancy between observed and expected counts, calculated as:
χ² = Σ [(O - E)² / E]
where O is the observed count and E is the expected count for each cell. Summing this value over all cells provides the test statistic. Calculating this by hand involves identifying each observed and expected pair, computing the squared differences divided by expected, and summing these results.
Determining p-Value and Degrees of Freedom
The degrees of freedom (df) for a 2x2 table are (rows - 1) × (columns - 1) = (2 - 1) × (2 - 1) = 1. The p-value corresponds to the probability of observing a chi-square statistic as extreme or more so under the null hypothesis. Using standard chi-square distribution tables or software like Minitab facilitates precise p-value determination. A small p-value (typically
Minitab Confirmation
In Minitab, the test involves inputting the degrees of freedom and the calculated chi-square statistic, then analyzing the right tail probability. This step provides validation of manual computations. The output includes the exact p-value, essential for making informed conclusions.
Conclusion and Reasoning
Based on the chi-square test results, if the p-value is less than the significance level (commonly 0.05), we reject the null hypothesis, concluding there is a statistically significant association between gender and sleep apnea status. Conversely, a p-value greater than 0.05 indicates insufficient evidence to conclude dependency, implying independence. The implications are crucial for healthcare strategies, emphasizing whether gender-specific approaches are justified.
References
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